Two Hall-MHD solutions that share their low Fourier modes up to a time-dependent determining wavenumber converge to each other in L2 as time goes to infinity.
Local well-posedness of the Hall-MHD system in $H^s(\mathbb {R}^n)$ with $s>\frac n2$
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abstract
We establish local well-posedness of the Hall-magneto-hydrodynamics (Hall-MHD) system in the Sobolev space $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2$. The previously known local well-posedness space was $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2+1$. Thus the result presented here is an improvement.
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Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics
Two Hall-MHD solutions that share their low Fourier modes up to a time-dependent determining wavenumber converge to each other in L2 as time goes to infinity.